A differential k-form can be integrated on an -dimensional manifold. The basic example is an -form in the open unit ball in . Since is a top-dimensional form, it can be written and so
(1)
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where the integral is the Lebesgue integral.
On a manifold covered by coordinate charts , there is a partition of unity such that
1. has support in and
2. .
Then
(2)
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where the right-hand side is well-defined because each integration takes place in a coordinate chart. The integral of the -form is well-defined because, under a change of coordinates , the integral transforms according to the determinant of the Jacobian, while an -form pulls back by the determinant of the Jacobian. Hence,
(3)
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is the same integral in either coordinate chart.
For example, it is possible to integrate the 2-form
(4)
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on the sphere . Because a point has measure zero, it is enough to integrate on , which can be covered by stereographic projection . Since
(5)
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the pullback map of is
(6)
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the integral of on is
(7)
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Note that this computation is done more easily by Stokes' theorem, because .